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74       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

of V which converges to/e V with respect to the weak topology (i.e., is such
that lim fn(x) =f(x) for all x e E) is also said to be weakly convergent. On

n~>oo

the same principle, we shall speak of subsets of V being weakly closed (in V),
weakly compact, etc. A subset B of KE is said to be weakly bounded if all
the seminorms (12.15.2) are bounded on B, that is if sup \f(x)\ < +00 for

/6B

all x e E. We shall also use the phrases weakly continuous (resp. weakly
differ entiable, weakly analytic) or scalarly continuous (resp. scalarly differen-
tiable, scalarly analytic) in place of " continuous (resp. differentiate, analytic)
with respect to the weak topology."

If E is a locally convex space (over K = R or C), the vector space of con-
tinuous linear forms on E is called the dual of E, and is often denoted by E'.
If xf e E' is a continuous linear form on E, its value at a point x e E is often
denoted by the expression <x, x'> or <X, xy instead of x'(x). Since E' is a
vector subspace of KB, it is endowed with the weak topology defined by the
seminorms x't-*\(x, x'y\. For each xeE, the. function x'\—>(x,x'y is a
weakly continuous linear form on E'.

If E is a Frechet space and E0 a dense vector subspace of E, then it follows
from (12.9.4) and the principle of extension of identities (3.15.2) that the
mapping x't-tx' |E0 is an (algebraic) isomorphism of the dual E' of E onto
the dual EJ of E0 .

(12.15.3) Let E, F be two locally convex spaces, E' and F' their respective
duals, and u : E -> F a continuous linear mapping. Then y'\-+y' ° u is a linear
mapping ofF' into E' which is continuous with respect to the weak topologies.

If y' is a continuous linear form on F, then yr ° u is a continuous linear
form on E. Given x0 e E, there exists y0 e F such that <%0 , y' o w> =
O'os.jO for all yr e F', namely yQ = w(jc0); the continuity of the mapping
y'\-»yfou is therefore an immediate consequence of (12.14.11) and the
definition (12.15.2) of the seminorms which define the weak topology.

We write yr o u = tu(y')9 and the linear mapping lu : F' -> E' is called the
transpose of the continuous linear mapping u. Thus we have

(12.15.4)                            <«<*), /> = <*,'«

for all x e E and y' e F'. Clearly

(1 2.1 5.5)             'fa + u2) = \ + * u2 ,        \Xu) =A'*u

for any scalar 1, and for any continuous linear mappings ui, u2 of E into F. Ifble metrizable subspace of F such that /(E) n L is